In this paper the problem of multiple scattering in a planetary atmosphere, both with and without a diffusely reflecting bottom surface, is discussed. We assume that the atmospheric scattering is isotropic, with an albedo a, and that the ground surface reflects the radiation according to Lambert's law, with an albedo b. Sections 11-TV are devoted to the standard problem, i.e., to the problem of a finite atmosphere without a ground surface. After the exact expressions for single and double scattering have been obtained in Section II by direct integration, we show in Section III that an indirect method gives the same result in a more elegant way. This method is based on the use of Chandrasekhar's equations by which the in- tensities of the reflected and transmitted radiation can be expressed in terms of two functions, X(s) and Y(s). A new feature in the present discussion is that physical meanings are assigned to these func- tions. With these meanings the general relations are newly derived; and the expressions for single and double scattering are obtained in a simple manner. Section IV completes these calculations by adding an approximate expression for the third-order scattering. A numerical example shows that in this way an accuracy of 0.1 per cent is reached for an atmosphere with r = 0.1. Section V discusses the planet problem, i.e., the problem of an atmosphere with a bottom surface. The solution of this problem is expressed in a general way in terms of the solution of the standard prob- 1cm. The exact solution for the case of an isotropically scattering atmosphere involves the functions X(s) and Y(s) and their moments. Formulae suitable for the case of a thin atmosphere are also given and illustrated by a numerical example. A comprehensive discussion of the nw1 Itemat ical funcliins occurring in these calculations is given in the appendix. They are of three types: The functions E~(x) are the ordinary exponential integrals. The functions F~(b, x) are definite integrals, of which the integrand is the product of an E-function and an exponential function. The functions Gnm(X) and G~m(X) are definite integrals, of which the integrand is the product of two E-functions. Only the last type of functions cannot be. expressed in terms of known functions. For their calculation a new elementary function, the exponential integral of the second order, is introduced. Numerical values of all functions needed are given in four tables